Growth of finitely generated solvable groups and curvature of Riemannian manifolds

If a group Γ is generated by a finite subset 5, then one has the gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J. Milnor [9] shows that the asymptotic behaviour of gs does not depend on choice of finite generating set S c Γ, and that lower (resp. upper) bounds on the curvature of a riemannian manifold M result in upper (resp. lower) bounds on the growth function of π^M). The types of bounds on the growth function are